Odd and even symmetric prime constellations

Number theory is concerned with the properties of integers and the research in number theory has been conducted since antiquity. As an example, one can mention the Erathostenes Sieve which sifts out primes from a number sequence, first mentioned in [1]. Further prime research have been the focus of many researchers following Erathostenes as can be seen from this list of references [2, 3, 4, 5, 6, 7, 8]. One of the prominent number theorists of the 20th century was Ramanujan for whom Hardy attributes the following anecdote: “I had ridden in taxicab No. 1729, and remarked that to me the number seemed a rather dull one, and that I hoped it was not an unfavourable omen. ’No,‘ he reflected, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways. That is, 1729 = 1³ + 12³ = 9³ + 10³.” [9]. This shows that one of the areas of interest for number theorists is to find various facts about specific integers or combinations of integers. In the past, the establishment of these facts was limited by the extent to which computations could be performed. Today, powerful computers have made it possible to vastly extend the size and complexity of these computations.

One area of number theory research has been to find patterns (constellations) in the sequence of integers and of particular interest has been finding patterns of primes, see for example the article by Granville [10, 11]. Also, the question of whether there is an unbounded number of twin primes has been extensively studied by [12, 13]. Another area of interest has been prime constellations (i.e. particular patterns of primes). Specific examples have been studied by [14, 15, 16] and others. For example, the Green-Tao theorem [14] proves that arbitrary long arithmetic progressions of primes exist.

As noted above, computers have made it possible to perform experiments to validate theoretical results in number theory as well as experimentally suggest conjectures for new theorems [17, 18]. In Zhang et. al. [19], it is stated that “it is known that primes contain unusual patterns”. The paper further lists some of these patterns.

The patterns called prime constellations are summarized in [20]. These patterns include twin primes, cousin primes, prime triplets, quadruplets and 5 tuplets.

In this paper, some new patterns called symmetric prime constellations are defined that have certain symmetry properties. These constellations are theoretically and computationally explored and examples of these constellations are exhibited. However, a result like the theorem quoted above [14] is unlikely to be proven for the constellations discussed here for reasons to be discussed hereafter. The theoretical and computational results presented are still a reflection of the fact that primes have certain distributive properties [10, 21]. These results and examples could lead to further conjectures and new results as discussed by Polya [22]. In [23] emerging applications of number theory, where the results in this paper might have applications, are surveyed. These applications include some important areas such as cryptography, information security, DNA modelling, image compression, among others.

By a “constellation”, we mean a finite pattern of primes.

This paper is concerned with certain symmetric constellations of n primes embedded in an increasing sequence of m integers j₁,⋯, j_m where j₁ and j_m are primes and j_i+1 − j_i = 1, i = 1, ⋯, m − 1, there are exactly n primes in the sequence and if j_i is a prime of the constellation then so is j_m−i. Such symmetric constellations are called C(m,n) constellations. The extent of a C(m,n) constellation is |C(m,n)| = m = j_{m − j1} + 1.

In this paper, the existence of C(m,n) constellations are demonstrated for various values of n and m. For some values of n, it is shown that there are constellations with the smallest possible extent.

Note that given a sequence of consecutive integers two primes in the sequence are separated by an odd number of integers.

The symmetric constellations C(m,n) can be divided into two classes depending on n being even or odd. Let the sequence of integer of the constellation be j₁, ⋯, j_m.

• If n is odd the center of the constellation is a prime c = ⌊j_m/j₁⌋.

• If n is even. Then the center of the constellation is c = (j_m/ − j₁)/2+1 where c is now a composite integer. In this case, the base case for constellations is given by ⋯, c − 1, c, c + 1, ⋯ where c − 1 and c + 1 are are primes and c is an even integer. A second case is defined by a central constellation c−2, c−1, c, c+1, c+2 where c − 2 and c + 2 are primes and c is an odd integer. A third case is defined by a central constellation c − 3, c − 2, c − 1, c, c + 1, c + 2, c + 3 where c − 3 and c + 3 are primes and c is even.

2.2

Preliminaries

The following result is frequently used: given a list of k consecutive positive integers, exactly one of the integers is divisible by k. (For a proof see for example [24].)

The following two results are known and also proven here.

Theorem 1

The minimum separation between two double prime centers is greater than or equal to 6.

Proof

Let p₁, d₁, p₂, c₁, p₃, d₂, p₄ be two double primes where d₂ −d₁ = 4. Then, d₁ and d₂ are both divisible by 3 which is not possible. Hence d₂ − d₁ > 4. The sequence 11, 12, 13, 14, 15, 16, 17 provides an example where d₂ − d₁ = 6.

Theorem 2

The constellation (1) p1,d1,p2,e1,c1,e2,p3,d2,p4,e3,c2,e4,p5,d3,p6 {p_1},{d_1},{p_2},{e_1},{c_1},{e_2},{p_3},{d_2},{p_4},{e_3},{c_2},{e_4},{p_5},{d_3},{p_6} of three double primes is not possible provided p₁ > 5.

Proof

Consider p₂, e₁, c₁, e₂, p₃. Then, one of the numbers must be divisible by 5. p₂ and p₃ are prime. Then e₁ is not divisible by 5 since it would imply the p₄ was divisible by 5. c₁ is also not divisible by 5 since it would imply that p₆ would be divisible by 5. Similarly e₂ cannot be divisible by 5 since it would imply p₁ be divisible by 5. This proves the theorem since none of the 5 numbers can be divisible by 5 in this constellation.

Note that there is one case of the constellation in Eq. 1 with the list of primes being (5, 7, 11, 13, 17, 19).

As noted, the divisibility with respect to 2, 3, 5 is determined by the two double primes in the sequence p₁, d₁, p₂, e₁, c₁, e₂, p₃, d₂, p₄. Carrying these divisibilities to the following sequence e₃, c₂, e₄, c₃, e₅, c₄, e₆, c₅, e₇, c₆, e₈, c₇, e₉ leaves only c₃, c₆, c₇ unconstrained by the divisibilities of 2, 3, 5 and leave the possibilities of these entries being composite or primes. Constraining c₆, c₇ to be primes then provides the smallest possible constellation of a pair of double primes at a minimum distance followed by a double prime. The centers of the first pair of double primes are d₁ and d₂ and the center of the double constellation is c₁. From Table 1, it is clear that the third double prime would have a center at c₈. Hence the minimum distance possible between the constellation consisting of the two pairs of double primes and the next double prime must be greater than or equal to c₈ − c₁ = 15.

Table 1

sequence	p1	d1	p2	e1	c1	e2	p3	d2	p4	e3	c2
divisors/prime	prime	2, 3	prime	2	3, 5	2	prime	2, 3	prime	2, 5	3
sequence	e4	c3	e5	c4	e6	c5	e7	c6	e8	c7	e9
divisors/prime	2		2, 3	5	2	3	2		2, 3, 5		2

In Table 2, it is computationally shown that the minimum distance between the two pairs of double primes is an achievable 15, or equivalently that the minimum extent for such a constellation is 21 = c₇ − p₁ + 1 where p₁ and c₇ are respectively the smallest and the largest primes.

Table 2

The pair of double primes.	p1	p2	p3	p4
The next double primes.					c6	c7
sequence 1	11	13	17	19	29	31
sequence 2	18041	18043	18047	18049	18059	18061
sequence 3	97841	97843	97847	97849	97859	97861
sequence 4	165701	165703	165707	165709	165719	165721
sequence 5	392261	392263	392267	392269	392279	392281
sequence 6	663581	663583	663587	663589	663599	663601
sequence 7	1002341	1002343	1002347	1002349	1002359	1002361
sequence 8	1068701	1068703	1068707	1068709	1068719	1068721

For sequences 2, 3, 5, 6 and 7 the entry c₃ is a composite number and for sequences 1, 3 and 8 c₃ is prime. Note also the regularity of the last digits of the primes. This is due to c₁ having a factor of 5. There are 35 such constellations up to 10 < 7 consisting of two double primes at minimum distance between them followed by a third double prime at smallest distance from the first two double primes.

The question is now, can Table 1 be expanded so that symmetric constellations can be defined? A possible expansion is shown in the table in Figure 1 where an added entry called rel. index has been added for each entry. The rel. index is a number indicating the relative integer distance to the central entry e₈ in the table. In the new table, the original divisor constraints with respect to 2, 3, 5 of Table 1 have been expanded by the divisor 7 tested for the range of rel. indices −19 to −13 with the original divisors 2, 3, 5 resulting in c₃, c₆, c₇, c₁₀, c₁₂, c₁₃, c₁₇, c₁₉ being unspecified. Then the red arrows show that all the entries at the 7 rel. indices −19 to −13 cannot be a multiple of 7 hence a contradiction to the assumption that a symmetric table with the primes p₁, p₂, p₃, p₄, c₃, c₆, c₇, c₁₀, c₁₂, c₁₃, c₁₇, c₁₉, i.e. a 12-tuple, is impossible in this form.

Fig. 1

Expanded Table 1.

The possible symmetric constellations C(m,n) which are now considered are restricted by the following theorems.

Theorem 3

Consider a symmetric C(m,n) constellation where n is odd. Then the center of the constellation is a prime P and the constellation can only have primes at offsets k * 6,k = 1,⋯from P.

Proof

The only possibility for the symmetric distribution of the primes constrains the center of the constellation to be a prime P. Let the sequence of integers around P be denoted by e_i, where |i| = 1, 2, ⋯. Consider e₋₁, P, e₁. Then, this is a sequence of three consecutive integers hence one and only one element of the sequence is divisible by 3. Suppose this is e₁. Then, the symmetry requirement means that e₂ is also divisible by 3. Hence e₋₁ and e₁ are both divisible by 3, a contradiction to the original claim that only one of the numbers in the sequence e₋₁, P, e₁ is divisible by 3. Now consider the sequence e₋₅, c₋₄, e₋₃, c₋₂, e₋₁, P,e₁, c₂, e₃, c₄, e₅. The same argument as above implies that c₂ and c₋₂ must be divisible by 3. Since e₁, e₃, e₅ and e₋₁, e₋₃, e₋₅ are even the first possible prime would be e₆. In the tables in Figures 2 and 3, the rel. index denotes the relative offsets of the integers from the center prime P which is at rel. index 0. By extending the above results to a table it follows that the only possible indices that can be prime are at locations offset from P by 6 * k, |k| = 1, 2, ⋯ as shown in the tables in Figure 2 and 3 depending on whether rel. index −1 or rel. index +1 is divisible by 3. The possible symmetric constellation are the same in both cases.

Fig. 2

Divisibility relative to center prime - odd case - rel. index −1 divisible by 3.

Fig. 3

Divisibility relative to center prime - odd case - rel. index 1 divisible by 3.

Theorem 4

Consider a symmetric C(m,n) constellation where n is even. Then the center of the constellation is a composite number c and the constellation base case can only have primes at locations given in table in Figure 4. The even second case can only have primes at locations given in table in Figure 5.

Proof

Due to the symmetry requirement the center of the constellation would have to be a composite number c with index 0 as used in Figures 4 and 5 or not an index item. In the first case, the extent of the constellation is m and the smallest resp. largest primes are p₁ and p_m. Then m = 2w + 1 where w is the number of items the left, resp. right of c. For the second case m = 2w which contradicts the fact that there is an odd number of times between two primes.

The base case of the 3 entries around the prime center for the even case is therefore p_i−1, c_i, p_i+1. Clearly c_i is divisible by 2 and 3. The possible base cases of even symmetric constellations would therefore have to conform to the divisibilities given in Figure 4.

The second case of even symmetric n-tuple has the 5 entries around the prime center as p_i−2, c_i−1, c_i, c_i+1, p_i+2 where c₋₁ and c+1 are even. Consider c_i, c_i+1, p_i+1. Suppose c_i+1 is divisible by 3. This would imply that p_i−2 is divisible by 3, hence a contradiction. So, c_i is divisible by 3. The resulting divisibilities are given in Figure 5.

Note that in Figure 2, there are empty non-colored entries at rel. indices −26, −14, −2, 10, 22 as well as −20, −8, 4, 16 and for Figure 3 the empty entries are −22, −10, 2, 14, 26 and −16, −4, 8, 20. As the value of n is increased some of these entries may be primes which will not have corresponding symmetric primes hence the constellation might be non-symmetric.

Some simple examples illustrating these definitions are provided when discussing the double (two-tuple) and quadruple (four-tuple) prime constellations.

In this paper, symmetric prime constellations are considered for various values of m and n. Of particular interest are symmetric constellations having the smallest possible extent for a fixed value of n.

Since a double prime is a pair of primes p₁ and p₂ separated by a composite even number e₁ it is an even symmetric C(3, 2) constellation which clearly has a minimal extent 3.

As noted, a great deal of research has also focused on whether there is an infinite number of double primes or not, see for example [12, 25].

Fig. 4

Divisibility relative to composite center - even base case.

Fig. 5

Divisibility relative to composite center - even second case.

One might ask if there are non-symmetric and symmetric prime constellations where n = 3 of the form p₁, e₁, p₂, e₂, p₃, i.e. m = 5. Indeed, there is at least one example: 3, 4, 5, 6, 7 where 3, 5, 7 are primes and 4, 6 are both even and composite.

To show that there are no further constellations of this kind the simple result used in Theorem 3 is needed. That is, given a sequence of n consecutive positive integers one and only one of the integers is divisible by n. Therefore considering that p₁, e₁, p₂ is a sequence of three numbers at least one has to be divisible by 3. Since p₁ and p₂ are primes it means that e₁ is divisible by 3. In the same manner p₂ or p₃ has to be divisible by 3. This means that one of the other sequences e₁, p₂, e₂ and e₁, p₂, e₂ has two numbers divisible by 3, which is not possible. The only resolution is that one of p₁, p₂ or p₃ is composite hence the C(5, 3) constellation p₁, e₁, p₂, e₂, p₃ is not possible. The next triple constellations possible are therefore either case_1: p₁, e₁, p₂, e₂, c₁, e₃, p₃ where c₁ is a composite integer or case_2: p₁, e₁, c₁, e₂, p₂, e₃, p₃ where c₁ is a composite. In both cases, the constellations are non-symmetric.

The first few examples of case_1 constellations are the triple sets of primes: (5, 7, 11), (11, 13, 17), (17, 19, 23), (41, 43, 47), (101, 103, 107), (107, 109, 113) where the first four constellations overlap and the last two constellations overlap. The first few examples of case_2 constellations are the triple sets of primes: (7, 11, 13), (13, 17, 19), (37, 41, 43), (67, 71, 73), (97, 101, 103), (103, 107, 109) where the first two constellations and the last two constellations overlap. In each case parentheses are used to clearly distinguish the constellations. These constellations are not symmetric and hence do not fall into the general theme of this paper. Checking if they can form a basis for 6-tuples or 7-tuples results in one case that will be considered in the section on 6-tuples. There are no possible 7-tuples on the basis of these two cases due to Theorem 3.

The question now is whether symmetric triple constellations exist and if therefore there are triple minimal constellations. The first potential triple symmetric constellation would be the C(9, 3) constellation p₁, e₁, c₁, e₂, p₂, e₃, c₂, e₄, p₃ where c₁, c₂ and c₃ are composite. Divisibility properties for this constellation are given in Table 3. The sequence is read from left to right and top to bottom.

Then, both from the divisibility by 3 for the sequence e₂, p₂, e₃ and from Theorem 3 it follows that this is not a possible symmetric constellation. The next potential symmetric constellation with 3 primes is the C(13, 3) constellation p₁, e₁, c₁, e₂, c₂, e₃, p₂, e₄, c₃, e₅, c₄, e₆, p₃. The properties of this constellation are given in Table 4.

Since this does not contradict Theorem 3, it is a possible symmetric triple constellation. The first example of such a constellation is given by Table 5.

Further examples for the centers p₂ of triple primes of this kind are 157, 167, 257, 263, 273. There are 758163 such triple primes up to 10⁹ showing that they are relatively common. Having eliminated the possibilities of smaller symmetric constellations of triple primes it follows that this version of C(13, 3) is a minimal symmetric triple constellation.

Quadruple prime constellations were discussed in the context of a hierarchy of primes in [26, 27].

Quadruple primes might be formed from two double primes each having 2-centers d₁ and d₂, with d₁ < d₂. This would define an even base case for a quadruple prime if they are separated by 3 non-primes. It is the smallest distance possible between two double primes if d₁ ≥ 6 (assuming that the 2-centers are greater than 6) (see also Theorem 4). A symmetric quadruple prime constellation sequence would therefore be p₁, e₁, p₂, e₂, c, e₃, p₃, e₄, p₄ where e₁ = d₁, e₄ = d₂ and the centers of such quadruple primes are given by c = (d₂ + d₂)/2.

Let's consider e₂, c, e₃. One of the integers has to be divisible by 3. It cannot be e₂ since this would imply p₂ would be divisible by 3, a contradiction since p₄ is a prime. For e₃ being divisible by 5 it would imply p₁ being composite, a contradiction. Hence c is divisible by 3. Then consider p₂, e₂, c, e₃, p₃ and assume one of the numbers is divisible by 5. It cannot be e₃ since this would imply that p₁ is composite, assume p₁ > 5. Similarly for e₃. Since p₂ and p₃ are primes, it shows that c must have a factor of 5. Note also that both one of e₁, e₂, c, e₃ and e₂, c, e₃, e₄ must be divisible by 7. In Table 6, the divisibilities for these quadruple primes are shown assuming p₁ is a prime greater than 5.

Computational examples of symmetric quadruple prime constellations are provided in Table 7.

There are 4768 symmetric quadruple C(9, 3) primes up to 10⁸.

It is also interesting that the sequences of the lowest digit are 1, 3, 7, 9 starting with the second quadruple prime. Since no counterexample to this was found up to 10⁷, it is conjectured that this sequence of lowest digit holds for all quadruple primes starting from the second one. The conjecture was easily verified by considering the trailing 5 for the center and then counting forwards and backwards from the center. I.e. p₁ is four steps backwards from q hence counting down from 5 results in 1 and so on.

These quadruple constellations are also minimal symmetric quadruple prime constellations from the above discussion.

The most obvious choice for a quintuple symmetric prime constellation would be to simply change the center of the quadruple prime constellation in Table 6 to a prime. This would result in a sequence of 5 primes only separated by even integers which can be easily shown to be impossible. It is also not possible from consulting the tables in Figures 2 and 3.

For the next choices for a symmetric quintuple prime Theorem 3 is consulted. Hence the first possible constellation with 5 primes will be with primes at the first two possible locations shown in Figure 2 or 3. Consider therefore the sequence of consecutive integers starting with p₁ (2) p1,e1,c1,e2,c2,e3,p2,e4,c3,e5,c4,e6,p3,e7,c5,e8,c6,e9,p4,e10,c7,e11,c8,e12,p5 {p_1},{e_1},{c_1},{e_2},{c_2},{e_3},{p_2},{e_4},{c_3},{e_5},{c_4},{e_6},{p_3},{e_7},{c_5},{e_8},{c_6},{e_9},{p_4},{e_{10}},{c_7},{e_{11}},{c_8},{e_{12}},{p_5} where p₁,⋯, p₅ are the primes of the 5-tuple and p₃ the center. Divisibilities of this sequence are shown in Table 8.

Consider then the sub-sequence c₄, e₆, p₃, e₇, c₅. One of the numbers in the sequence must be divisible by 5. Assume it is c₄. This would imply p₁ is composite which is not possible since it is a prime. p₃ is prime by assumption. c₅ is not possible since it would imply p₅ was composite. In a similar manner e₆ and e₇ cannot be divisible by 5. Hence there is a contradiction since one of the 5 consecutive numbers must be divisible by 5.

For the next possible constellations expanded the constellations by adding p₁ − 6 and p₅ + 6 to be prime at rel. indices −18, 18 (Case_1) and then choosing either the entries at rel. location −6, 6 to be prime or rel. entries locations −12, 12 to be prime (Case_2).

For case 1, it results in Table 9 where the constellation entries have been renamed.

However, consider the sequence e₉, p₃, e₁₀. One of these integers must be divisible by 3. Suppose this is e₉. Then, this would mean that p₂ is composite hence a contradiction. A similar argument holds for e₁₀. Hence, the Case_1 constellation is not possible.

The second choice is to allow the original p₂ and p₄ to be composite and require the entries at rel. index −6 and +6 to be prime. The resulting table with renamed entries and divisibilities is Table 10 and where p₃ is still the center prime of the constellation.

Case_2 was tested numerically and the following results were obtained as shown in Table 11.

There are 5162 instances of these quintuple primes up to 10⁹. It is interesting to note that there are two types of these primes depending on the sequence of low order digits. These types are distinguished by the divisibility by 3 of the even integers that are just before or just after the center prime.

In the same manner, as in the previous section an obvious choice for symmetric six-tuple constellations would be to start with the symmetric quadruple constellations and extend those with a prime at the first extension from the extremes of the quadruple constellation obtaining a new constellation p₁, e₁, p₂, e₂, p₃, e₃, c₁, e₄, p₄, e₅, p₅, e₆, p₆.

This is clearly not a possible constellation since it would include sequences of three consecutive primes (i.e. p₁, e₁, p₂, e₂, p₃) only separated by even integers which has been shown to be impossible, except for the trivial case of 3, 4, 5.

Hence, either p₁ and p₂ (and symmetrically p₅ and p₆) or p₂ and p₃ (and symmetrically p₄ and p₅) have to be separated by an even-odd-even sequence.

In the first case, the resulting sequence would be (3) p1,e1,c1,e2,p2,e3,p3,e4,c2,e5,p4,e6,p5,e7,c3,e8,p6. {p_1},{e_1},{c_1},{e_2},{p_2},{e_3},{p_3},{e_4},{c_2},{e_5},{p_4},{e_6},{p_5},{e_7},{c_3},{e_8},{p_6}.

Let's Consider e₄, c₂, e₅. One of the integers in this sequence must be divisible by 3. Suppose this is e₄. This would imply p₄ being divisible by 3 which is not possible since it is a prime. A similar argument holds for e₅. Hence c₂ is divisible by 3.

Now consider p₃, e₄, c₂, e₅, p₄. In this case, one of the integers in the sequence must be divisible by 5. Let's Consider e₄. It is is divisible by 5 then this implies p₅ is divisible by 5 which is not possible since p₅ is a prime. By symmetry e₅ is not divisible by 5. Since p₃ and p₄ are primes it leaves c₂ which is therefore divisible by 5.

For the sequence p₃, e₄, c₂, e₅, p₄, e₆, p₅ consider again e₄. If e₄ is divisible by 7 then so would be p₁ as well, which is not possible since p₁ is a prime. Similarly e₅ is excluded from being divisible by 7. For e₆ being divisible by 7 would imply p₂ begin divisible by 7, yet another contradiction. The remaining integers are prime, except for c₂ which therefore is divisible by 7.

The resulting divisibility properties of the sequence in Eq. 3 are provided in Table 12. This is an example of an even constellation that conforms to the divisibility properties found in Figure 5.

There are two items of interest in this case. First of all, the low order digits of the primes in the constellations form a sequence 7, 11, 13, 17, 19, 23 by subtracting the first prime less 7 from the sequence of primes for a given sixtuple. That is, for example for the second sixtuple the same sequence as is given for the first sixtuple prime is repeated if 97 − 7 is subtracted from the primes. This is explained by considering that c₂ in Table 12 is divisible by 5 and not divisible by 2, hence the low digit of c₂ is 5. The rest follows by counting up and down from c₂.

There is also a large jump from the first 5 sixtuples to the following ones as shown in Table 13.

There are 18 sixtuples of this kind up to 10⁷.

One can ask if it is possible to combine two triple prime constellations symmetrically around an even center to obtain sixtulpe primes and if so, do they have a smaller width. The divisibility of such a constellation would be as in Table 14. That is, the sequence p₃, e₇, p₄ constants of three consecutive integers, one of which has to be divisible by 3. Since e₇ is the only possible choice the entries divisible by 3 can be entered into the table together with the divisibility by 2. Then consider the sequence c₄, e₆, p₃, e₇, p₄ of consecutive integers. One of these integers must be divisible by 5. It cannot be c₄ since this would imply that p₁ is a prime. Similarly e₆ is not a candidate since this would imply that p₂ would be a prime. Since p₃ and p₄ are primes it leaves e₇ which is therefore divisible by 5.

The first few symmetric sixtuples of this kind are given in Table 15

There are 66 such sixtuples up to 10⁷.

There is also another similar case as shown in Table 16. In previous case there was one composite center with a prime on each side. In the new case there are two primes at the center of the sixtuple separated by three composite numbers. Otherwise the two cases are similar. Here c₅ is clearly divisible by 3. Then consider the sequence p₃, e₇, c₅, e₈, p₄. One of these integers must be divisible by 5. Since e₇ cannot be divisible by 5 since it would imply p₆ being composite it follows that c₅ is the only choice due to the symmetry and the fact that p₃ and p₄ are prime.

There are 34 sixtuples of this kind up to 10⁷ and the first 7 examples are shown in Table 17. Note that the last digits of the primes are the same for each sixtuple due to the center being odd and divisible by 5.

Considering the table in Figure 5 the question is whether the minimal possible configuration p₁, e₁, p₂, e₂, c₁, e₃, p₃, e₄, p₄, e₅, c₂, e₆, p₅, e₇, p₆ in Table 18 will produce sixtuple primes.

A computation up to 10⁹ produced one example: 5, 7, 11, 13, 17, 19.

Proof for the impossibility of such constellations for p₁ > 5 was given in Theorem 2.

From the table in Figure 5 it can be seen that another possible minimal sixtuple constellation in this case would be p₁, e₁, c₁, e₂, p₂, e₃, p₃, e₄, c₂, e₅, p₄, e₆, p₅, e₇, c₃, e₈, p₆. Will this C(17, 6) constellation produce symmetric sixtuple prime constellations?

Again a computation produces one example 5, 9, 11, 15, 17, 21. A proof similar to the proof in Theorem 2 shows that there are no further examples when p₁ > 5.

From Theorem 3, it is clear that the only possible symmetric seventuple prime constellations have to involve primes which are offset k * 6, k = 1⋯ steps away from the prime center of the constellation.

In the following the rel. index will denote the locations of primes symmetrically relative to a prime center c. As an example, the locations of the primes in the case of −24, −12, −6, 0, 6, 12, 24 are as shown in Eq. 4. (4) ⋯, −24, ⋯, −12, ⋯, −6, ⋯, c,⋯, 6, ⋯, 12, ⋯, 24,⋯ \cdots ,\, - 24,\, \cdots ,\, - 12,\, \cdots ,\, - 6,\, \cdots ,\,c, \cdots ,\,6,\, \cdots ,\,12,\, \cdots ,\,24, \cdots

Also, from the discussion above it turns out that the first offset with the first prime greater than 5 is not a possible choice as noted when discussing the quintuple constellations. Hence the prime locations of Eq. 4 are not possible.

Numerically experimenting with the choice 1 −24, −18, −12, 0, 12, 18, 24 provided no seventuples for values up to 10⁹. The possible proof that there are no seventuples of this form is missing currently.

The next possible choice was therefore offsets −30, −18, −12, 0, 12, 18, 30 with the center prime p₄. In Table 19, the sequence of primes is shown interleaved between composite even and composite odd numbers. The divisibility with respect to 2, 3 in this case is then shown in Table 20 for the case when the even number to the left of the center prime p₄ is divisible by 3. The divisibility of the case where the number to the right of the center is divisible by 3 is symmetric with respect to the center to the first case and not displayed here.

The first few seventuples of the type shown in Tables 19 and 20 are given in Table 21 and there are 73 such seventuples up to 10⁹.

Note the number of blank entries for the divisibilities in Table 20. These entries can be composite or prime. If one or more of these entries are prime then the constellation might not satisfy the symmetry condition and it would have more than 7 primes. However, if the two entries c₁₀ and c₁₅ are the added primes then the resulting constellation is symmetric and might possibly be a 9-tuple. Referring to Figure 6 this possibility is now contradicted shown as follows. The red boxes are the original 7-tuple primes and yellow boxes the two added possible primes. The green rectangle have 5 consecutive entries and hence one should be a multiple of 5. Then the red arrows point to p₅ and p₆ which are a multiple of 5 steps from e₁₄ and c₁₂ respectively. The yellow arrows point to c₁₀ and c₁₅ which are 5 steps each from e₁₅ and e₁₆. The last item is p₄ which is a prime. Since none of the entries in the green box can be a multiple of 5 it contradicts the original assumption and hence the constellation is not possible.

Fig. 6

Proof of impossibility of suggestion for Table 20.

In general, as the tuples increase in number these added non-specified entries cause greater and greater problems when searching for pure n-tuple constellations, especially for the odd numbered tuples, where by pure is meant that there are no other primes in the constellation.

In [26, 27], general octuple prime constellations with centers n + 19 were discussed.

These octuple constellations were generated on the bases of two quadruple constellations based on one 4-tuple with entries p₁, e₁, p₂, e₂, c₁, e₃, p₃, e₄, p₄ and the first possible following 4-tuple with entries p₅, e₁₆, p₆, e₁₇, c₁₂, e₁₈, p₇, e₁₉, p₈, e₂₀, as shown in Table 22. In the general case the entries at c₆ and c₇ were allowed to be primes or composite numbers. Only the cases where both of those entries are composite numbers confirm to the strict definition used here for symmetric prime constellations. The widths of these octal constellations are m = 39.

Note that the number of undetermined entries is 2 in this case whereas in the 7-tuple case there were 14 such entries (Table 20).

Examples of these symmetric prime constellations were computed up to 210 * 10⁹ finding 439 such constellations.

Another possibility is to consider Table 12 and then asking that c₁ and c₃ be primes. This would result in two double primes that are too close, hence disproving the existence of such constellations.

A further approach to symmetric octuple constellations is to start with the sequence prime-composite-prime and then check to see if this sequence can be the central core of an octal constellation.

Computing with this C(39, 8) constellation resulted in 29 cases of 8-tuple up to 10⁹. As examples, the primes of the first and last case are given in Table 24.

Yet another possibility is to start with a 6-tuple constellation as shown in Table 14. Noting that in this case the entries c₁ and c₈ are required to be composite for the 6-tuple case it seems reasonable to ask for what happens if these entries are required to be prime. Testing this up to 10¹⁰ results in 102 cases of eigthtuple primes where the first few are given in Table 25.

A further example of octal primes is provided by the sequence of rel. indices (−17, −13, −11, −1, 1, 11, 13, 17)

This results in 160 case where the first few have centers: (30, 103980, 113160, 1322160, 2668230). Of these cases 28 have a prime at rel. index −7 or 7 or both.

An example of octal primes with a larger span is given by the case C(47, 8) with rel. indices (−23, −19, −17, −1, 1, 17, 19, 23). When testing for examples 66 cases were found up to 10¹⁰ with 24 of the cases having one or more of integers with rel. indices (−13, −11, −7, 7, 11, 13) being prime.

A second example with C(47, 8) is provided by rel. indices (−23, −19, −13, −1, 13, 19, 23). Up to 30 * 10¹⁰ 199 examples were found. 89 of the examples had one or more of the integers at re. indices (−17, −11, −7, 7, 11, 17) being prime.

A third example with C(47, 8) with rel. indices of the primes being (−23, −19, −11, −1, 1, 11, 19, 23) provided 90 examples of which 40 had one or more primes at the rel. indices (−17, −13, −7, 7, 13, 17).

A further example C(47, 8) constellations with rel. indices of the primes being (−23, −13, −7, −1, 1, 7, 13, 23) provided 115 examples of which 60 had one or more primes at the rel. indices (−19, −17, −11, 11, 17, 19).

As noted when the number of the primes increases for the odd symmetry case the number of possible constellations are reduced.

This lead to a conjecture that there were no symmetric 9-tuple constellations possible. The conjecture was disproved by selecting the sequence of primes at rel. indices (−60, −42, −30, 0, 18, 30, 42, 60).

Checking for 9-tuple primes up to 2 * 10¹⁰ one case was found as shown in Table 26.

A further larger number of examples with one or more added primes were also found which did not satisfy the strict symmetry requirements, but some of these could possibly be 12-tuples.

For some 10-tuple prime constellations consider Table 23 and the sequence of rel. indices (−19, −17, −13, −11, −7, −1, 1, 7, 11, 13, 17, 19) in the table. As noted, it was shown that for this case all of the rel. indices could not be prime. Removing a pair of rel. indices might, however, provide examples for 10-tuples.

For the first case of changing the entries at rel. indices −19, 19 to be composite two examples of 10-tuples with rel. indices (−17, −13, −11, −7, −1, 1, 7, 11, 13, 17) were found when testing up to 10¹⁰ as shown in Table 27.

For the second case of rel. indices the three (−19, −17, −13, −11, −1, 1, 11, 13, 17, 19) allowing the entries at rel. indices (−7, 7) to be composite the three cases listed in Table 28.

For the third case of rel. indices (−19, −17, −13, −11, −7, 7, 11, 13, 17, 19) allowing the entries at rel. indices −1, 1 to be composite the four cases listed in Table 29 were found.

In Table 22 the entries c₆ and c₇ were not specified (see also Figure 5 from [26]). If these additional entries were primes then this would be examples of symmetric C(39, 10) of even case one prime constellations.

Numerically testing for this case resulted in three examples of such constellations up to 210 * 10⁹. These constellations had centers 39713433690, 66419473050, 71525244630. In Table 30 the center of the first 10-tuple prime is t = 39713433690 = 2 * 3 * 5 * 7 * 23 * 8222243.

In Figure 7 the complete list of integers and their divisors are given for the first prime listed in Table 30.

Fig. 7

Complete table for the first 10-tuple.

Some examples pf 10-tuple symmetric constellations C(47, 10) are now provided.

The first example is provided by the sequence of primes at rel/index (−23, −17, −13, −7, −1, 1, 7, 13, 17, 23)

Testing up to 30 * 10¹⁰ resulted in one example with the rel. index center 30.

For the sequence of primes at rel/index (−23, −19, −17, −13, −1, 1, 13, 17, 19, 23) no results were obtained when testing for centers up to 30 * 10¹⁰.

Another example of C(47, 10) is provided by the sequence of primes at rel. index (−23, −17, −13, −11, −1, 1, 11, 13, 17, 23). In this case there were 3 resulting 10-tuples with centers 30, 807213960, 6296082240. The result with the center 30 has three extra primes at the integers 11, 23, 37 whereas the other two results had no extra primes in the ranges of the constellations..

Another example C(47, 10) is provided by the sequence of primes at rel. index (−23, −19, −17, −13, −11, 11, 13, 17, 19, 23) where one result was found with the central prime being 6697423110 = (2, 3, 5, 7, 167, 353, 541).

A further example C(47, 10) is provided by the sequence of primes at rel. index (−23, −17, −13, −7, −1, 1, 7, 13, 17, 23) One case with center at 30 when testing for examples up to 30 * 10¹⁰.

An example where the rel. indices of the 10-tuple are (−23, −19, −17, −11, −7, 7, 11, 17, 19, 23) results in 5 cases when testing up to 30 * 10¹⁰ one of which has an added prime at rel. index 3. The centers of the 5 case are listed in Table 31.

For a last example the rel. indices are (−23, −19, −13, −7, −1, 1, 7, 13, 19, 23). Of the three examples found with centers listed in Table 32, the first constellation has primes at re. indices (−17, 11) and the second constellation has a prime at rel index (11). The last case has no primes at the rel. indices (−17, −11, 11, 17).

Table 33 lists the minimal even symmetric constellations found so far.

Table 34 lists the moinimal odd symmetric constellations found so far.

Symmetric prime constellations have been defined and a number of cases of such constellations have been found numerically.

There are a number of possible further questions that can be explored.

• Is there any relationship between and n-tuple and the minima extent of the n-tuple?

• What is the maximal n for an odd n-tuple and for an even n-tuple?

• What are the conditions for two n-tuple forming a 2n-tuple for even n and a 2n + 1-tuple for odd n (assuming the center of the 2n + 1-tuple is a prime).

• Can a symmetric 12-tuple constellation be found?

• For each value of n can all the possible constellations up to a specified limit be found?

• Does a 16-tuple exist as based on the octuples from the hierarchy presented in [26]?

If this would be the case, such a 16-tuple would have to conform to the divisibilities shown in Figure 8.

Fig. 8

Table for a possible 16-tuple based on octuples.